Applicable Algebra in Engineering Communication and Computing最新文献

In this paper, we compute sequence covering arrays (SCAs), which are arrays, consisting of sequences, such that all subsequences with pairwise different entries of some length are covered, via a novel approach based on commutative algebra and symbolic computation. Hereby, we provide various algebraic models being capable to characterize possibly small sets of permutations collectively containing particular shorter subsequences. These models take the form of multivariate polynomial systems of equations and are then processed via supercomputing by a Gröbner Basis solver in order to compute solutions from them. If the variety is not empty, i.e. the Gröbner basis is non-trivial, then each point in the computed variety can be transformed to a SCA. In our experiments, we observed varying computational performance depending on the chosen model, while all of them exhibited scalability issues. Additionally and for comparison, we give new SAT descriptions modelling SCAs. By employing a SAT solver on our provided SAT models, we are able to provide upper bounds, one of which is best among literature results. Lastly, we adapt our SAT approach to answer a question posed by Yuster (Des Codes Cryptogr 88(3):585–593, 2020). As a result, we find a characterization of the dimensions of all perfect SCAs with coverage multiplicity two of strength three.

The choice of the elliptic curve for a given pairing based protocol is primordial. For many cryptosystems based on pairings such as group signatures and their variants (EPID, anonymous attestation, etc) or accumulators, operations in the first pairing group (mathbb {G}) of points of the elliptic curve is more predominant. At 128-bit security level two curves BW13-P310 and BW19-P286 with odd embedding degrees 13 and 19 suitable for super optimal pairing have been recommended for such pairing based protocols. But a prime embedding degree ((k=13;19)) eliminates some important optimisation for the pairing computation. However The Miller loop length of the superoptimal pairing is the half of that of the optimal ate pairing but involve more exponentiations that affect its efficiency. In this work, we successfully develop methods and construct algorithms to efficiently evaluate and avoid heavy exponentiations that affect the efficiency of the superoptimal pairing. This leads to the definition of new bilinear and non degenerate pairing on BW13-P310 and BW19-P286 called x-superoptimal pairing where its Miller loop is about (15.3 %) and (39.8 %) faster than the one of the optimal ate pairing previously computed on BW13-P310 and BW19-P286 respectively.

Let (pequiv 1pmod 4) be a prime. In this paper, we support a new method, i.e., a product of 2-adic values for four binary sequences, to determine the maximum evaluations of the 2-adic complexity in all almost balanced cyclotomic binary sequences of order four with period (N=p), which are viewed as generalizing the results in Hu (IEEE Trans. Inf. Theory 60:5803–5804, 2014) and Xiong et al. (IEEE Trans. Inf. Theory 60:2399–2406, 2014) without the autocorrelation values of cyclotomic binary sequences of order four with period p. By number theory we obtain two necessary and sufficient conditions about the 2-adic complexity of all balanced cyclotomic binary sequences of order four with period (N=2p) and show the 2-adic complexity of several non-balanced cyclotomic binary sequences of order four with period 2p, which are viewed as generalizing the results in Zhang et al. (IEEE Trans. Inf. Theory 66:4613–4620, 2020).

A linear code C of length (n = ru + sv) is a two-dimensional ({mathbb {F}})-double cyclic code if the set of coordinates can be partitioned into two arrays, such that any cyclic row-shifts and column-shifts of both arrays of a codeword is also a codeword. In this paper, we examine the algebraic structure of these codes and their dual codes in general. Moreover, we are interested in finding out a generating set for these codes (and their dual codes) in case when (u=2), (v=4) and char((F) ne 2).

It is known that the following five counting problems lead to the same integer sequence ({f_t}({n})):

1. (1)

   the number of nonequivalent compact Huffman codes of length n over an alphabet of t letters,

2. (2)

   the number of “nonequivalent” complete rooted t-ary trees (level-greedy trees) with n leaves,

3. (3)

   the number of “proper” words (in the sense of Even and Lempel),

4. (4)

   the number of bounded degree sequences (in the sense of Komlós, Moser, and Nemetz), and

5. (5)

   the number of ways of writing

   $$begin{aligned} 1= frac{1}{t^{x_1}}+ dots + frac{1}{t^{x_n}} end{aligned}$$

   with integers (0 le x_1 le x_2 le dots le x_n).

In this work, we show that one can compute this sequence for all (n<N) with essentially one power series division. In total we need at most (N^{1+varepsilon }) additions and multiplications of integers of cN bits (for a positive constant (c<1) depending on t only) or (N^{2+varepsilon }) bit operations, respectively, for any (varepsilon >0). This improves an earlier bound by Even and Lempel who needed ({O}({{N^3}})) operations in the integer ring or (O({N^4})) bit operations, respectively.

The projective norm graphs are central objects to extremal combinatorics. They appear in a variety of contexts, most importantly they provide tight constructions for the Turán number of complete bipartite graphs (K_{t,s}) with (s>(t-1)!). In this note we deepen their understanding further by determining their automorphism group.

In 2016, Y. Kumar et al. in the paper ‘Affine equivalence and non-linearity of permutations over ({mathbb {Z}}_n)’ conjectured that: For (nge 3), the nonlinearity of any permutation on ({mathbb {Z}}_n), the ring of integers modulo n, cannot exceed (n-2). For an odd prime p, we settle the above conjecture when (n=2p) and for (pequiv 3 pmod {4}) we prove the above conjecture with an improved upper bound. Further, we derive a lower bound on (max {mathcal {N}}{mathcal {L}}_n) when n is an odd prime or twice of an odd prime where (max {mathcal {N}}{mathcal {L}}_n) denotes the maximum possible nonlinearity of any permutation on ({mathbb {Z}}_n).

In this work, we give three new techniques for constructing Hermitian self-dual codes over commutative Frobenius rings with a non-trivial involutory automorphism using (lambda)-circulant matrices. The new constructions are derived as modifications of various well-known circulant constructions of self-dual codes. Applying these constructions together with the building-up construction, we construct many new best known quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40.

Let p be a prime and n be a positive integer. We consider rational functions (f_b(X)=X+1/(X^p-X+b)) over ({mathbb {F}}_{p^n}) with (textrm{Tr}(b)ne 0). In Hou and Sze (Finite Fields Appl 68, Paper No. 10175, 2020), it is shown that (f_b(X)) is not a permutation for (p>3) and (nge 5), while it is for (p=2,3) and (nge 1). It is conjectured that (f_b(X)) is also not a permutation for (p>3) and (n=3,4), which was recently proved sufficiently large primes in Bartoli and Hou (Finite Fields Appl 76, Paper No. 101904, 2021). In this note, we give a new proof for the fact that (f_b(X)) is not a permutation for (p>3) and (nge 5). With this proof, we also show the existence of many elements (bin {mathbb {F}}_{p^n}) for which (f_b(X)) is not a permutation for (n=3,4).